An equivariant analog of the Poincare-Hopf theorem
by
Konstantin Feldman
Moscow State University

We introduce a new concept of the index of the zero of the tangent vector field on the smooth manifold. This concept generalizes the usual index of the non-degenerated isolated zero of the smooth tangent vector field. On the basis of this concept we prove an equivariant analog of the classical Poincare-Hopf theorem. As an application of the main result we deduce an exact addition formula for the first-half integer Pontryagin class in complex cobordism.

Let us consider a tangent vector field s on a smooth manifold M. We denote the connected component of the zero set of s as M₁. We assume that M₁ is a closed compact submanifold of M. There is such a deformation of s in an open neighborhood N(M₁) of M₁, that the vector field s defines a map:

s|N(M1): \nu(M1) --> ~ \tau   (\nu(M1)),

where \nu(M₁) is a normal bundle of an inclusion M₁ subset M, [(\tau)\tilde](\nu(M₁)) a tangent bundle along the fiber. Note that [(\tau)\tilde](\nu(M₁)) =~ p^*\nu(M₁), where p:\nu(M₁) --> M₁ is a projection. We define the produced bundle map as [p\tilde]:[(\tau)\tilde](\nu(M₁)) --> \nu(M₁). There is such a vector bundle \zeta that \nu(M₁)\oplus\zeta is a trivial vector bundle.

Definition 1. The index of the zero set M₁ of the vector field s is an element of \pi⁰_S(M⁺₁). The element is defined by the equivalent class of the following map:

j:SN /\ M+1 = T(\nu(M1)\oplus\zeta) ( ~ p   , id) -->   T(\nu(M1)\oplus\zeta) \pi -->   SN.

Let \tau(p) be the transfer map for the fibre bundle (E, F, B, p) with the closed compact smooth fibre F without boundary. We assume that there is a vector field s, tangent to E along the fibre. Let E₁, ..., E_m be the irreducible components of the zeros of the vector field.

Theorem 1. The stable homotopical equality holds

{\tau(p)}= m å j=1  i*j({Inds(Ej)}{\tau(pj)}),

where \tau(p_j) is the transfer map for E_j, i_j:E_j --> E is an inclusion.

We fix a cogomology theory h^*(·) and assume that all bundles E, E₁, ..., E_m are h-orientated in addition to the conditions of theorem 1.

Corollary 1. We fix w in h^*(E). Suppose that there is an element \Delta in h^*(E), which does not divide zero, and the Euler class of the tangent bundle to E along the fibre divides p^*(\Delta)w. Then the following decomposition formula for the Gysin map holds

p!(w)= m å j=1  pj! \frac Inds(Ej)i*j(w)e(\nuj),

where e(\nu_j) is the Euler class of the normal bundle E_j subset E in h^*(·).

Theorem 2. There are such elements \beta_ij in \Omega_U that for the first half-integer Pontryagin class in complex cobordism the addition formula holds:

p1/2(\xi\oplus\zeta)=u+v+ å i, j >= 1  \betaij si-1(u)sj-1(v),

where u=p_1/2(\xi), v=p_1/2(\zeta), s_m are the Landveber-Novikov operations.

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V.M.Buchstaber, K.E.Feldman, The index of the equivariant vector field and the addition theorems for the Pontryagin characteristic classes, (to appear in Izv. RAS).

Date received: June 2, 1999